An inequality involving the second largest and smallest eigenvalue of a distance-regular graph

2010/04/07 by Jack H. Koolen, Koolen, Jack H., Jongyook Park +3 · 1 citation
Mathematics · Computer Science · #Finite Group Theory Research #Graph theory and applications #Coding theory and cryptography

paper · pdf · doi:10.48550/arxiv.1004.1056

Abstract

For a distance-regular graph with second largest eigenvalue (resp. smallest eigenvalue) μ1 (resp. \muD) we show that (μ1+1)(\muD+1)<= -b1 holds, where equality only holds when the diameter equals two. Using this inequality we study distance-regular graphs with fixed second largest eigenvalue.

Cited by

• Examples of diameter-2 graphs with no triangle or K_2,t

Related

• The distance-regular graphs such that all of its second largest local eigenvalues are at most one
• Distance-regular graphs with a few q-distance eigenvalues
• Distance-regular graphs with valency k having smallest eigenvalue at most -k/2
• Non-geometric distance-regular graphs of diameter at least 3 with smallest eigenvalue at least -3
• Non-bipartite distance-regular graphs with a small smallest eigenvalue